Page 318 -

§2. Generalities on -Adic Representations 293

(1.2) Proposition. The divisor A F is well-defined by E in the divisor class group
Cl(R) of R, and as a divisor it satisfies the relation

12A F + ( F ) = D E = v( v )v
v
In the divisor class group D E is 12A F .
If 12 kills the divisor class group Cl(R) of divisors modulo principal divisors,
then D E equals (d E ) the divisor of a function. The class group Cl(R) is zero if and
only if R is a principal ideal ring, and in this case we have a globally minimal normal
cubic equation for an elliptic curve. The following theorem was already seen from
the discussion before 5(2.6), but it also follows from the discussion above.

(1.3) Theorem. Let R be a principal ideal ring, and let E be an elliptic curve over
the field of fractions K. There exists a normal cubic equation F E (x, y) = 0 for E
over K such that F = F E is minimal at all places v of R. Further, d E equals F up
to a unit in R.

Proof. Since A F = (u) for some nonzero u in K, we can use u to define an isomor-
phism of the curve with equation F = 0 onto the curve with A F = 0, that is, we can
make a change of variable to the case A F = 0. Then, for such an equation F over R,
we have

( f ) = D E = (d E ).

This proves the theorem.
We conclude this section with some notations concerning algebraic number
fields.
(1.4) Notations. Let K denote an algebraic number field with ring of integers R
and for each place v of K, we denote the completion of K at v by K v .If v is
Archimedean, then K v is isomorphic to R or C. For a non-Archimedean valuation
v, let R v (resp. R (v) ) denote the valuation ring of v in K v (resp. K) consisting of all
field elements a with v(a) ≥ 0. Let π v denote a local uniformizing parameter of v
which can be chosen to be in K satisfying v(π v ) = 1, and let k(v) denote the residue
class field of v. Since R v π v (resp. R (v) π v ) is the unique maximal ideal of R v (resp.
R (v) ), the residue class field k(v) is either R v /R v π v or R (v) /R (v) π v . Finally, let Nv
a
be the order of k(v) and p v the characteristic of k(v) where p = Nv.
v

§2. Generalities on -Adic Representations

For a field k with separable algebraic closure k s , the Galois group Gal(k s /k) is given
the Krull topology where a neighborhood base of the identity consists of subgroups
fixing finite extensions K of k. With this topology, it is a compact and totally discon-
nected group.